Pendulums & Simple Harmonic Motion

The steady swing that keeps time

Junior level — plain language, no maths

Tie a weight to a string, pull it to one side and let go: it swings across, back, across, back — over and over, in a rhythm so even you could set a clock by it. People did exactly that for three hundred years. That swinging weight is a pendulum, and the steadiness of its beat is one of the quiet marvels of physics.

The surprising part is what the rhythm depends on — and what it doesn't. Make the weight heavier: the timing doesn't change. Pull it out further so it swings wider: it travels faster but takes the same time to get back, so the timing barely changes either. The one thing that really matters is the length of the string. Short string, quick busy ticks; long string, slow lazy swings. Legend says Galileo spotted this as a boy, timing a swaying lamp in a cathedral against his own pulse.

That reliability made the pendulum the heart of the most accurate clocks in the world for centuries. And the same back-and-forth rhythm is everywhere once you look: a child on a swing, a guitar string humming, a spring bouncing, even a skyscraper swaying in the wind. In the simulation below, change the length and the swing size and watch what happens to the beat.

Simple harmonic motion and the pendulum period

Student level — the core equations

Nudge a pendulum off centre and gravity pulls it back — and crucially, for small swings the restoring pull is very nearly proportional to the displacement, \(F \approx -kx\). Any system obeying that one rule performs simple harmonic motion: the position traces a perfect sine wave in time, \(x(t) = A\cos(\omega t)\), with angular frequency \(\omega = \sqrt{k/m}\). The same equation governs a mass on a spring, a floating buoy, and the vibration of a molecule.

For a pendulum the maths delivers the famous result \(T = 2\pi\sqrt{L/g}\): the period grows with the square root of the length and shrinks in stronger gravity — and contains neither the mass nor the amplitude. That is the pendulum's isochronism: quadruple the length to double the period; swap the bob for one twice as heavy and nothing changes. The frequency is just \(f = 1/T\), the number of swings per second.

Energy is what keeps it going. At the ends of the swing all the energy is potential (the bob is highest, momentarily still); at the bottom it is all kinetic (lowest and fastest), and the total \(E = \tfrac{1}{2}kA^2\) stays fixed as it sloshes endlessly between the two. Strictly, all this holds only for small angles — swing too wide and the restoring force stops being perfectly linear, and the period creeps a little longer.

Nonlinearity, damping, resonance and chaos

Scholar level — full mathematical depth

01Why the small-angle trick works

The true equation of a pendulum is nonlinear: \(\ddot\theta + (g/L)\sin\theta = 0\). It has no solution in elementary functions. The escape route is the approximation \(\sin\theta \approx \theta\), valid to within 1% out to about 14°, which linearises the equation into the harmonic oscillator \(\ddot\theta + \omega^2\theta = 0\) with \(\omega = \sqrt{g/L}\). Almost all of "simple" harmonic motion is really this quiet decision to throw away the higher terms of a sine.

02What the approximation costs

Keep the full \(\sin\theta\) and the period stops being constant — it lengthens with amplitude, exactly, as a complete elliptic integral \(T = 4\sqrt{L/g}\,K(\sin\tfrac{\theta_0}{2})\). Expanded, \(T \approx T_0\left(1 + \tfrac{1}{16}\theta_0^2 + \cdots\right)\): a 30° swing runs about 1.7% slow, a 90° swing nearly 18%. Isochronism is a small-angle fiction — real pendulum clocks are engineered to keep the amplitude tiny and constant for exactly this reason.

03Adding friction: the damped oscillator

Reality drains energy. A resistive force proportional to speed gives \(\ddot x + 2\gamma\dot x + \omega_0^2 x = 0\), whose behaviour splits three ways by the damping ratio \(\zeta\): underdamped (\(\zeta<1\)) rings down with a decaying envelope \(e^{-\gamma t}\); critically damped (\(\zeta=1\)) returns to rest fastest without overshoot — the target for a car suspension or a door closer; overdamped (\(\zeta>1\)) crawls back sluggishly.

04Pushing at the right moment: resonance

Drive an oscillator with a periodic force and its steady response peaks sharply when the drive frequency approaches the natural \(\omega_0\) — resonance, amplitude \(A(\omega) = F_0/m\big/\sqrt{(\omega_0^2-\omega^2)^2 + 4\gamma^2\omega^2}\). Push a swing in time and it climbs higher and higher on tiny nudges. The same effect shatters a wine glass with a sung note, and it is why soldiers break step crossing a bridge.

05How sharp is the peak: the Q factor

The quality factor \(Q = \omega_0/2\gamma\) measures how lightly damped an oscillator is — the number of radians it rings through before its energy falls by \(1/e\), and the sharpness of its resonance. A car suspension has \(Q \sim 1\); a quartz watch crystal reaches \(10^4\text{–}10^6\); the mirrors of the LIGO gravitational-wave detector are engineered toward \(Q \sim 10^7\), which is what lets them register a length change of \(10^{-18}\) m.

06Two rods, no prediction: chaos

Hang one pendulum from the end of another and the tidy world collapses. The double pendulum is fully deterministic yet chaotic: its motion depends so sensitively on the starting angle that two releases differing by a hair diverge completely within seconds, a positive Lyapunov exponent in action. Its trajectory in phase space fills a strange, folded region rather than closing into a neat loop — the same mathematics that makes the weather unforecastable beyond a couple of weeks.